Measurement in Scientific Study and Uncertainty in Measurement

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Measurement in Scientific Study
andUncertainty in Measurement
Lecture 3

• Chemistry 142 B
• James B. Callis, Instructor
• Autumn Quarter, 2004

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Precision and Accuracy Errors in
Scientific Measurements
Precision - Refers to reproducibility or how
close the measurements are to
each other. Accuracy - Refers to how close a
measurement is to the true
value. Systematic Error - produces values that
are either all higher
or all lower than the actual value. Random
Error - in the absence of systematic error,
produces some
values that are higher and some that
are lower than the actual value.
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Rules for Determining Which Digits Are
Significant
All digits are significant, except zeros that are
used only to position the decimal point.
1. Make sure that the measured quantity has a
decimal point. 2. Start at the left of the number
and move right until you reach the first
nonzero digit. 3. Count that digit and every
digit to its right as significant. Zeros that end
a number and lie either after or before the
decimal point are significant thus 1.030 mL
has four significant figures, and 5300. L has
four significant figures also. Numbers such as
5300 L have 2 sig. figs., but 5.30x103 L has 3.
A terminal decimal point is often used to clarify
the situation, but scientific notation is clearer
(best).
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Examples of Significant Digits in Numbers
Number - Sig digits Number
- Sig digits
0.0050 L 1.34000 x
107 nm six 18.00 g four
5600 ng 0.00012 kg two
87,000 L two 83.0001 L
six 78,002.3 ng
six 0.006002 g four
0.000007800 g four 875,000 oz
1.089 x 106 L 30,000 kg one
0.0000010048 oz five 5.0000 m3
five 6.67000 kg
six 23001.00 lbs seven 2.70879000
mL nine 0.000108 g 1.0008000 kg
eight 1,470,000 L three
1,000,000,000 g
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Examples of Significant Digits in Numbers
Number - Sig digits Number
- Sig digits
0.0050 L two 1.34000 x 107
nm six 18.00 g four
5600 ng two 0.00012 kg
two 87,000 L
two 83.0001 L five
78,002.3 ng six 0.006002 g
four 0.000007800 g
four 875,000 oz three 1.089 x
10 -6L four 30,000 kg one
0.0000010048 oz five 5.0000 m3
five 6.67000 kg
six 23,001.00 lbs seven 2.70879000
mL nine 0.000108 g three
1.0008000 kg eight 1,470,000 L
three 1,000,000,000 g one
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Rules for Significant Figures in answers
1. For multiplication and division. The number
with the least certainty limits the certainty of
the result. therefore, the answer contains the
same number of significant figures as there are
in the measurement with the fewest significant
figures. Multiply the following numbers
9.2 cm x 6.8 cm x 0.3744 cm
2. For addition and subtraction. The answer has
the same number of decimal places as there are
in the measurement with the fewest decimal
places. Example, adding two volumes 83.5 mL
23.28 mL Example subtracting two volumes
865.9 mL - 2.8121393 mL
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Rules for Significant Figures in answers
1. For multiplication and division. The number
with the least certainty limits the certainty of
the result. therefore, the answer contains the
same number of significant figures as there are
in the measurement with the fewest significant
figures. Multiply the following numbers
9.2 cm x 6.8 cm x 0.3744 cm 23.4225 cm3 23 cm3
2. For addition and subtraction. The answer has
the same number of decimal places as there are
in the measurement with the fewest decimal
places. Example, adding two volumes 83.5 mL
23.28 mL 106.78 mL 106.8 mL Example
subtracting two volumes 865.9 mL - 2.8121393
mL 863.0878607 mL 863.1 mL
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Rules for Rounding Off Numbers

• (1) In a series of calculations, carry the extra
  digits through to the final result, then round
  off.
• (2) If the digit to be removed
• is less than 5, the preceding digit stays the
  same. For example, 1.33 rounds to 1.3.
• is equal to or greater than 5, the preceding
  digit is increased by one. For example, 1.36
  rounds to 1.4.
• (3) When rounding, use only the first number to
  the right of the last significant figure. Do not
  round off sequentially. For example, the number
  4.348 when rounded to two significant figures is
  4.3, not 4.4.
• Notes
• Your TI-93 calculator has the round function
  which you can use to get the correct result. Find
  round by pressing the math key and moving to NUM.
  Its use is round(num, no of decimal places
  desired), e.g. round(2.746,1) 2.7.
• Your book will show intermediate results
  rounded off. Dont use these rounded results to
  get the final answer.

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Rounding Off Numbers Problems
(3-1a) Round 5.379 to three significant figures
Ans (3-1b) Round 5.379 to two significant
figures Ans We used the rule If the digit
removed is greater than or equal to 5, the
preceding number increases by 1. (3-2a) Round
0.2413 to three significant figures Ans (3-2b)
Round 0.2413 to two significant figures Ans We
used the rule If the digit removed is less than
5, the preceding number is unchanged
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Sample Problem 3-3
Lithium (Li) is a soft, gray solid that has the
lowest density of any metal. If a slab of Li
weighs 1.49 x 103 mg and has sides that measure
20.9 mm by 11.1 mm by 12.0 mm, what is the
density of Li in g/ cm3 ?
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Sample Problem 3-3(cont.)
Mass (g) of Li 1.49 x 103 mg Length (cm) of
one side 20.9 mm Similarly, the other side
lengths are Volume (cm3) Density
mass/volume Density of Li
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Problem 3-4 Volume by Displacement
Problem Calculate the density of an irregularly
shaped metal object that has a mass of 567.85
g if, when it is placed into a 2.00 liter
graduated cylinder containing 900.00 mL of
water, the final volume of the water in the
cylinder is 1277.56 mL ? Plan Calculate the
volume from the different volume readings,
and calculate the density using the mass that
was given. Solution
Volume
mass
Density
volume
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Definitions - Mass Weight
Mass - The quantity of matter an object contains
kilogram - ( kg ) - the SI base unit of mass, is
a platinum -
iridium cylinder kept in
Paris as a standard!
Weight - depends upon an objects mass and the
strength of the gravitational
field pulling on it, i.e. w f ma.
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Problem 3-5 Computer Chips
Future computers might use memory bits which
require an area of a square with 0.25 mm sides.
(a) How many bits could be put on a 1 in x 1 in
computer chip? (b) If each bit required that 25
of its area to be coated with a gold film 10 nm
thick, what mass of gold would be needed to make
one chip?

• Approach
• use Achip
• (b) use r m/V

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Solution to Chip Problem (3-7)
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Solution to Chip Problem (3-7)
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Temperature Scales and Interconversions
Kelvin ( K ) - The Absolute temperature scale
begins at absolute zero
and only has positive values.
Celsius ( oC ) - The temperature scale used by
science, formally
called centigrade and most
commonly used scale around the world,
water freezes at 0oC, and boils
at 100oC.
Fahrenheit ( oF ) - Commonly used scale in
America for our
weather reports, water freezes at 32oF,
and boils at 212oF.
T (in K) T (in oC) 273.15 T (in oC) T (in
K) - 273.15
T (in oF) 9/5 T (in oC) 32 T (in oC) T
(in oF) - 32 5/9
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Problem 3-6Temperature Conversions
(a) The boiling point of Liquid Nitrogen is
-195.8 oC, what is the temperature in Kelvin and
degrees Fahrenheit?
T (in K) T (in oC) 273.15 T (in K)
T (in oF) 9/5 T (in oC) 32 T (in oF)
(b)The normal body temperature is 98.6oF, what is
it in Kelvin and degrees Celsius?
T (in oC) T (in oF) - 32 5/9 T (in oC)
T (in K) T (in oC) 273.15 T (in K)
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Answers to Problems in Lecture 3

1. (a)5.38 (b) 5.4
2. (a) 0.241 (b) 0.24
3. 0.536 g/cm3
4. 1.5040 g / mL
5. 31 mg gold
6. (a) 77.4 K -320.4 oF (b) 37.0 oC 310.2 K